Related papers: Groups without unitary representations, submeasure…

Let $(U, R)$ be an approximation space with $U$ being non-empty set and $R$ being an equivalence relation on $U$, and let $\overline{G}$ and $\underline{G}$ be the upper approximation and the lower approximation of subset $G$ of $U$. A…

We show that, for a generic measure preserving transformation $T$, the closed group generated by $T$ is not isomorphic to the topological group $L^0(\lambda, {\mathbb T})$ of all Lebesgue measurable functions from $[0,1]$ to $\mathbb T$…

Let $G$ be a real reductive Lie group, $L$ a compact subgroup, and $\pi$ an irreducible admissible representation of $G$. In this article we prove a necessary and sufficient condition for the finiteness of the multiplicities of $L$-types…

Ruberman in the 90's showed that the group of exotic diffeomorphisms of closed 4-manifolds can be infinitely generated. We provide various results on the question of when such infinite generation can localize to a smaller embedded…

The Evasiveness conjecture have been proved for properties of graphs on a prime-power number of vertices and the six vertices case. The ten vertices case is still unsolved. In this paper we study the size of the automorphism group of a…

A reduction $\varphi$ of an ordered group $(G,P)$ to another ordered group is an order homomorphism which maps each interval $[1,p]$ bijectively onto $[1, \varphi(p)]$. We show that if $(G,P)$ is weakly quasi-lattice ordered and reduces to…

We prove that finite entropy random walks on the torsion-free Baumslag group in dimension $d=2$ have non-trivial Poisson boundary. This is in contrast with the torsion case where the situation for simple random walks on Baumslag groups is…

For a broad class of symplectic manifolds of dimension at least six, we find the following new phenomenon: there exist local exotic Lagrangian tori. More specifically, let $X$ be a geometrically bounded symplectic manifold of dimension at…

It has been known since the time of Nielsen that the mapping class group $\text{Mod}_{g,1}$ of a surface of genus $g$ and one puncture acts faithfully by homeomorphisms on the circle. In this note, we show that this standard representation…

Arthur classified the discrete automorphic representations of symplectic and orthogonal groups over a number field by that of the general linear groups. In this classification, those that are not from endoscopic lifting correspond to pairs…

We prove that given a finite rank free group $\mathbb{F}$ of rank $\geq 3$ and two exponentially growing outer automorphisms $\psi$ and $\phi$ with dual lamination pairs $\Lambda^\pm_\psi$ and $\Lambda^\pm_\phi$ associated to them, and…

The question studied here is the behavior of the Poisson bracket under C^0-perturbations. In this purpose, we introduce the notion of pseudo-representation and prove that for a normed Lie algebra, it converges to a representation. An…

Let $G$ be a finite group and $\varphi(G)=|\{a\in G \mid o(a)=\exp(G)\}|$, where $o(a)$ denotes the order of $a$ in $G$ and $\exp(G)$ denotes the exponent of $G$. Under a natural hypothesis, in this note we determine the groups $G$ such…

We show that the Baum-Connes morphism twisted by a non-unitary representation, defined in [GA08], is an isomorphism for a large class of groups satisfying the Baum-Connes conjecture. Such class contains all the real semi-simple Lie groups,…

We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups,…

We prove that every sofic approximation of a property (T) group is approximately isomorphic to one having geometric property (T), and more generally, a box space of graphs which has boundary geometric property (T) is approximately…

Let $\phi: G \rightarrow H$ be a group homomorphism such that $H$ is a totally disconnected locally compact (t.d.l.c.) group and the image of $\phi$ is dense. We show that all such homomorphisms arise as completions of $G$ with respect to…

A 1930s conjecture of Hopf states that an even-dimensional compact Riemannian manifold with positive sectional curvature has positive Euler characteristic. We prove this conjecture under the additional assumption that the isometry group has…

We show that if $H \leq G$ is a closed amenable and cocompact subgroup of a unimodular locally compact group, then the reduced group C*-algebra of $G$ is not simple. Equivalently, there are unitary representations of $G$ that are weakly…

We show that an accessible group with infinitely many ends has property $R_{\infty}$. That is, it has infinitely many twisted conjugacy classes for any twisting automorphism. We deduce that having property $R_{\infty}$ is undecidable…